Nonlinear supersymmetry in Quantum Mechanics: algebraic properties and differential representation

TL;DR

This paper investigates the algebraic structure of nonlinear supersymmetry in one-dimensional quantum mechanics, revealing polynomial relations, limitations on supercharges, and hidden symmetries within the superalgebra.

Contribution

It provides a detailed analysis of the polynomial structure of nonlinear SUSY algebras, especially for the case of two supercharges, and introduces the concept of hidden symmetry operators.

Findings

SUSY algebra with transposition symmetry is polynomial in the Hamiltonian.

No more than two independent supercharges generate a nonlinear superalgebra.

Existence of a non-trivial hidden symmetry operator as a nonlinear function of the Super-Hamiltonian.

Abstract

We study the Nonlinear (Polynomial, N-fold,...) Supersymmetry algebra in one-dimensional QM. Its structure is determined by the type of conjugation operation (Hermitian conjugation or transposition) and described with the help of the Super-Hamiltonian projection on the zero-mode subspace of a supercharge. We show that the SUSY algebra with transposition symmetry is always polynomial in the Hamiltonian if supercharges represent differential operators of finite order. The appearance of the extended SUSY with several (complex or real) supercharges is analyzed in details and it is established that no more than two independent supercharges may generate a Nonlinear superalgebra which can be appropriately specified as {\cal N} = 2 SUSY. In this case we find a non-trivial hidden symmetry operator and rephrase it as a non-linear function of the Super-Hamiltonian on the physical state space. The full {\cal N} = 2 Non-linear SUSY algebra includes "central charges" both polynomial and non-polynomial (due to a symmetry operator) in the Super-Hamiltonian.